Massimiliano Gubinelli and Martina Hofmanová
We prove existence of global solutions to singular SPDEs on $\mathbb{R}^d$ with cubic nonlinearities and additive white noise perturbation, both in the elliptic setting in dimensions $d=4,5$ and in the parabolic setting for $d=2,3$. We prove uniqueness and coming down from infinity for the parabolic equations. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the $\Phi^4_d$ Euclidean quantum field theory via Parisi–Wu stochastic quantization, while the elliptic equations are linked to the $\Phi^4_{d-2}$ Euclidean quantum field theory via the Parisi–Sourlas dimensional reduction mechanism.
Massimiliano Gubinelli and Martina Hofmanová. Global solutions to elliptic and parabolic $\Phi^4$ models in Euclidean space. Papers of Massimiliano Gubinelli, 2018, version 2. https://mgubi.org/papers/paper/phi4-global-solutions/
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